Problem: What is $\left(\frac{7}{8}\right)^3 \cdot \left(\frac{7}{8}\right)^{-3}$?
By definition, if $a$ is nonzero, then $a^{-3}$ is the reciprocal of $a^3$.  So, $\left(\frac78\right)^3$ and $\left(\frac78\right)^{-3}$ are reciprocals.  Therefore, their product is $\boxed{1}$.